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\section{Partial differential equations}
\subsection{Definition}
In an ordinary differential equation (ODE) the infinitesimal change of the dependent variable (i.e the function) depends on the rate of change of only one independent variable (e.g time). In turn, in a partial differential equation (PDE), the dependent variable changes with respect to two or more variables (for example time and space). PDEs are often used to describe dynamic processes in multidimensional systems (e.g processes that change in both time and space), like heat conduction, voltage propagation along conductive cables or calcium diffusion within defined volumes.

As with ODEs, the solution to a PDE is simply a function or a family of functions. For PDEs, the solution depends not only on one independent variable, but on various independent variables.


\subsection{Partial derivatives}
If we have a function $f$ whose indepedent variable are $x,y,z,t$, the total derivative of the function $f$ with respect to $t$ is

\begin{equation*}
{d \over dt} f(x,y,z,t) = 
    \frac{\partial f}{\partial t}  +
    \frac{\partial f}{\partial x}{dx \over dt} +
    \frac{\partial f}{\partial y}{dy \over dt} +
    \frac{\partial f}{\partial z}{dz \over dt}
\end{equation*}
